DAMASK_EICMD/code/constitutive_j2.f90

! Copyright 2011 Max-Planck-Institut für Eisenforschung GmbH
!
! This file is part of DAMASK,
! the Düsseldorf Advanced MAterial Simulation Kit.
!
! DAMASK is free software: you can redistribute it and/or modify
! it under the terms of the GNU General Public License as published by
! the Free Software Foundation, either version 3 of the License, or
! (at your option) any later version.
!
! DAMASK is distributed in the hope that it will be useful,
! but WITHOUT ANY WARRANTY; without even the implied warranty of
! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
! GNU General Public License for more details.
!
! You should have received a copy of the GNU General Public License
! along with DAMASK. If not, see <http://www.gnu.org/licenses/>.
!
!##############################################################
!* $Id$
!*****************************************************
!*      Module: CONSTITUTIVE_J2                      *
!*****************************************************
!* contains:                                         *
!* - constitutive equations                          *
!* - parameters definition                           *
!*****************************************************

! [Alu]
! plasticity              j2
! (output)                flowstress
! (output)                strainrate
! c11                     110.9e9    # (3 C11 + 2 C12 + 2 C44) / 5  ... with C44 = C11-C12 !!
! c12                     58.34e9    # (1 C11 + 4 C12 - 1 C44) / 5
! taylorfactor            3
! tau0                    31e6
! gdot0                   0.001
! n                       20
! h0                      75e6
! tausat                  63e6
! a                       2.25

module constitutive_j2

 use prec, only: pReal,pInt
 
 implicit none
 private
 character (len=*), parameter, public :: constitutive_j2_label = 'j2'
 
 integer(pInt),   dimension(:), allocatable, public :: &
   constitutive_j2_sizeDotState, &
   constitutive_j2_sizeState, &
   constitutive_j2_sizePostResults
   
 integer(pInt), dimension(:,:), allocatable, target, public :: &
   constitutive_j2_sizePostResult     ! size of each post result output
   
 character(len=64), dimension(:,:), allocatable, target, public :: &
   constitutive_j2_output             ! name of each post result output
 
 integer(pInt), dimension(:),  allocatable, private :: &
   constitutive_j2_Noutput                                                               !< name of each post result output

 character(len=32), dimension(:), allocatable, private :: &
   constitutive_j2_structureName
   
 real(pReal), dimension(:), allocatable, private :: &
 !* Visco-plastic constitutive_j2 parameters
   constitutive_j2_fTaylor, &
   constitutive_j2_tau0, &
   constitutive_j2_gdot0, &
   constitutive_j2_n, &
 !* h0 as function of h0 = A + B log (gammadot) 
   constitutive_j2_h0, &
   constitutive_j2_h0_slopeLnRate, &
   constitutive_j2_tausat, &
   constitutive_j2_a, &
   constitutive_j2_aTolResistance, &
 !* Parameters of normalized strain rate vs. stress function:
 !* tausat += (asinh((gammadot / SinhFitA)**(1 / SinhFitD)))**(1 / SinhFitC) / (SinhFitB * (gammadot / gammadot0)**(1/n))
   constitutive_j2_tausat_SinhFitA, &
   constitutive_j2_tausat_SinhFitB, &
   constitutive_j2_tausat_SinhFitC, &
   constitutive_j2_tausat_SinhFitD

 real(pReal), dimension(:,:,:), allocatable, private :: &
   constitutive_j2_Cslip_66

 public  :: constitutive_j2_init, &
            constitutive_j2_stateInit, &
            constitutive_j2_aTolState, &
            constitutive_j2_homogenizedC, &
            constitutive_j2_microstructure, &
            constitutive_j2_LpAndItsTangent, &
            constitutive_j2_dotState, &
            constitutive_j2_deltaState, &
            constitutive_j2_dotTemperature, &
            constitutive_j2_postResults

contains

subroutine constitutive_j2_init(myFile)
!**************************************
!*      Module initialization         *
!**************************************
 use, intrinsic :: iso_fortran_env                                ! to get compiler_version and compiler_options (at least for gfortran 4.6 at the moment)
 use math, only: &
   math_Mandel3333to66, &
   math_Voigt66to3333
 use IO, only: &
   IO_lc, &
   IO_getTag, &
   IO_isBlank, &
   IO_stringPos, &
   IO_stringValue, &
   IO_floatValue, &
   IO_error
 use material
 use debug, only: &
   debug_level, &
   debug_constitutive, &
   debug_levelBasic
 use lattice, only: lattice_symmetrizeC66  

 implicit none
 integer(pInt), intent(in) :: myFile
 
 integer(pInt), parameter :: maxNchunks = 7_pInt
 
 integer(pInt), dimension(1_pInt+2_pInt*maxNchunks) :: positions
 integer(pInt) :: section = 0_pInt, maxNinstance, i,o, mySize
 character(len=64)   :: tag
 character(len=1024) :: line = ''                                                                   ! to start initialized

 write(6,*)
 write(6,*) '<<<+-  constitutive_',trim(constitutive_j2_label),' init  -+>>>'
 write(6,*) '$Id$'
#include "compilation_info.f90"
 
 maxNinstance = int(count(phase_plasticity == constitutive_j2_label),pInt)
 if (maxNinstance == 0_pInt) return

 if (iand(debug_level(debug_constitutive),debug_levelBasic) /= 0_pInt) then
   write(6,'(a16,1x,i5)') '# instances:',maxNinstance
   write(6,*)
 endif
 
 allocate(constitutive_j2_sizeDotState(maxNinstance))
          constitutive_j2_sizeDotState = 0_pInt
 allocate(constitutive_j2_sizeState(maxNinstance))
          constitutive_j2_sizeState = 0_pInt
 allocate(constitutive_j2_sizePostResults(maxNinstance))
          constitutive_j2_sizePostResults = 0_pInt
 allocate(constitutive_j2_sizePostResult(maxval(phase_Noutput), maxNinstance))
          constitutive_j2_sizePostResult = 0_pInt
 allocate(constitutive_j2_output(maxval(phase_Noutput), maxNinstance))
          constitutive_j2_output = ''
 allocate(constitutive_j2_Noutput(maxNinstance))
          constitutive_j2_Noutput = 0_pInt
 allocate(constitutive_j2_structureName(maxNinstance))
          constitutive_j2_structureName        = ''
 allocate(constitutive_j2_Cslip_66(6,6,maxNinstance))
          constitutive_j2_Cslip_66 = 0.0_pReal
 allocate(constitutive_j2_fTaylor(maxNinstance))
          constitutive_j2_fTaylor = 0.0_pReal
 allocate(constitutive_j2_tau0(maxNinstance))
          constitutive_j2_tau0 = 0.0_pReal
 allocate(constitutive_j2_gdot0(maxNinstance))
          constitutive_j2_gdot0 = 0.0_pReal
 allocate(constitutive_j2_n(maxNinstance))
          constitutive_j2_n = 0.0_pReal
 allocate(constitutive_j2_h0(maxNinstance))
          constitutive_j2_h0 = 0.0_pReal
 allocate(constitutive_j2_h0_slopeLnRate(maxNinstance))
          constitutive_j2_h0_slopeLnRate = 0.0_pReal
 allocate(constitutive_j2_tausat(maxNinstance))
          constitutive_j2_tausat = 0.0_pReal
 allocate(constitutive_j2_a(maxNinstance))
          constitutive_j2_a = 0.0_pReal
 allocate(constitutive_j2_aTolResistance(maxNinstance))
          constitutive_j2_aTolResistance = 0.0_pReal
 allocate(constitutive_j2_tausat_SinhFitA(maxNinstance))
          constitutive_j2_tausat_SinhFitA = 0.0_pReal
 allocate(constitutive_j2_tausat_SinhFitB(maxNinstance))
          constitutive_j2_tausat_SinhFitB = 0.0_pReal
 allocate(constitutive_j2_tausat_SinhFitC(maxNinstance))
          constitutive_j2_tausat_SinhFitC = 0.0_pReal
 allocate(constitutive_j2_tausat_SinhFitD(maxNinstance))
          constitutive_j2_tausat_SinhFitD = 0.0_pReal
 
 rewind(myFile)
 
 do while (IO_lc(IO_getTag(line,'<','>')) /= 'phase')                                                                              ! wind forward to <phase>
   read(myFile,'(a1024)',END=100) line
 enddo
 
 do                                                                                                                                ! read thru sections of phase part
   read(myFile,'(a1024)',END=100) line
   if (IO_isBlank(line)) cycle                                                                                                     ! skip empty lines
   if (IO_getTag(line,'<','>') /= '') exit                                                                                         ! stop at next part
   if (IO_getTag(line,'[',']') /= '') then                                                                                         ! next section
     section = section + 1_pInt                                                                                                    ! advance section counter
     cycle
   endif
   if (section > 0_pInt .and. phase_plasticity(section) == constitutive_j2_label) then                                             ! one of my sections
     i = phase_plasticityInstance(section)                                                                                         ! which instance of my plasticity is present phase
     positions = IO_stringPos(line,maxNchunks)
     tag = IO_lc(IO_stringValue(line,positions,1_pInt))                                                                            ! extract key
     select case(tag)
       case ('plasticity','elasticity')
         cycle
       case ('(output)')
         constitutive_j2_Noutput(i) = constitutive_j2_Noutput(i) + 1_pInt
         constitutive_j2_output(constitutive_j2_Noutput(i),i) = IO_lc(IO_stringValue(line,positions,2_pInt))
       case ('lattice_structure')
         constitutive_j2_structureName(i) = IO_lc(IO_stringValue(line,positions,2_pInt))
       case ('c11')
              constitutive_j2_Cslip_66(1,1,i) = IO_floatValue(line,positions,2_pInt)
       case ('c12')
              constitutive_j2_Cslip_66(1,2,i) = IO_floatValue(line,positions,2_pInt)
       case ('c13')
              constitutive_j2_Cslip_66(1,3,i) = IO_floatValue(line,positions,2_pInt)
       case ('c22')
              constitutive_j2_Cslip_66(2,2,i) = IO_floatValue(line,positions,2_pInt)
       case ('c23')
              constitutive_j2_Cslip_66(2,3,i) = IO_floatValue(line,positions,2_pInt)
       case ('c33')
              constitutive_j2_Cslip_66(3,3,i) = IO_floatValue(line,positions,2_pInt)
       case ('c44')
              constitutive_j2_Cslip_66(4,4,i) = IO_floatValue(line,positions,2_pInt)
       case ('c55')
              constitutive_j2_Cslip_66(5,5,i) = IO_floatValue(line,positions,2_pInt)
       case ('c66')
              constitutive_j2_Cslip_66(6,6,i) = IO_floatValue(line,positions,2_pInt)
       case ('tau0')
              constitutive_j2_tau0(i) = IO_floatValue(line,positions,2_pInt)
       case ('gdot0')
              constitutive_j2_gdot0(i) = IO_floatValue(line,positions,2_pInt)
       case ('n')
              constitutive_j2_n(i) = IO_floatValue(line,positions,2_pInt)
       case ('h0')
              constitutive_j2_h0(i) = IO_floatValue(line,positions,2_pInt)
       case ('h0_slope','slopelnrate')
              constitutive_j2_h0_slopeLnRate(i) = IO_floatValue(line,positions,2)
       case ('tausat')
              constitutive_j2_tausat(i) = IO_floatValue(line,positions,2_pInt)
       case ('tausat_sinhfita')
              constitutive_j2_tausat_SinhFitA(i) = IO_floatValue(line,positions,2)
       case ('tausat_sinhfitb')
              constitutive_j2_tausat_SinhFitB(i) = IO_floatValue(line,positions,2)
       case ('tausat_sinhfitc')
              constitutive_j2_tausat_SinhFitC(i) = IO_floatValue(line,positions,2)
       case ('tausat_sinhfitd')
              constitutive_j2_tausat_SinhFitD(i) = IO_floatValue(line,positions,2)
       case ('a', 'w0')
              constitutive_j2_a(i) = IO_floatValue(line,positions,2_pInt)
       case ('taylorfactor')
              constitutive_j2_fTaylor(i) = IO_floatValue(line,positions,2_pInt)
       case ('atol_resistance')
              constitutive_j2_aTolResistance(i) = IO_floatValue(line,positions,2_pInt)
       case default
              call IO_error(210_pInt,ext_msg=tag//' ('//constitutive_j2_label//')')
     end select
   endif
 enddo

100 do i = 1_pInt,maxNinstance                                        ! sanity checks
   if (constitutive_j2_structureName(i) == '')            call IO_error(205_pInt,e=i)
   if (constitutive_j2_tau0(i) < 0.0_pReal)               call IO_error(211_pInt,ext_msg='tau0 (' &
                                                               //constitutive_j2_label//')')
   if (constitutive_j2_gdot0(i) <= 0.0_pReal)             call IO_error(211_pInt,ext_msg='gdot0 (' &
                                                               //constitutive_j2_label//')')
   if (constitutive_j2_n(i) <= 0.0_pReal)                 call IO_error(211_pInt,ext_msg='n (' &
                                                               //constitutive_j2_label//')')
   if (constitutive_j2_tausat(i) <= 0.0_pReal)            call IO_error(211_pInt,ext_msg='tausat (' &
                                                               //constitutive_j2_label//')')
   if (constitutive_j2_a(i) <= 0.0_pReal)                 call IO_error(211_pInt,ext_msg='a (' &
                                                               //constitutive_j2_label//')')
   if (constitutive_j2_fTaylor(i) <= 0.0_pReal)           call IO_error(211_pInt,ext_msg='taylorfactor (' &
                                                               //constitutive_j2_label//')')
   if (constitutive_j2_aTolResistance(i) <= 0.0_pReal)    call IO_error(211_pInt,ext_msg='aTol_resistance (' &
                                                               //constitutive_j2_label//')')
 enddo

 do i = 1_pInt,maxNinstance
   do o = 1_pInt,constitutive_j2_Noutput(i)
     select case(constitutive_j2_output(o,i))
       case('flowstress')
         mySize = 1_pInt
       case('strainrate')
         mySize = 1_pInt
       case default
         call IO_error(212_pInt,ext_msg=constitutive_j2_output(o,i)//' ('//constitutive_j2_label//')')
     end select
  
     if (mySize > 0_pInt) then                               ! any meaningful output found
       constitutive_j2_sizePostResult(o,i) = mySize
       constitutive_j2_sizePostResults(i) = &
       constitutive_j2_sizePostResults(i) + mySize
     endif
   enddo

   constitutive_j2_sizeDotState(i) = 1_pInt
   constitutive_j2_sizeState(i)    = 1_pInt
   
   constitutive_j2_Cslip_66(:,:,i) = lattice_symmetrizeC66(constitutive_j2_structureName(i),&
                                                                      constitutive_j2_Cslip_66(:,:,i)) 
   constitutive_j2_Cslip_66(1:6,1:6,i) = &
     math_Mandel3333to66(math_Voigt66to3333(constitutive_j2_Cslip_66(1:6,1:6,i)))

 enddo

end subroutine constitutive_j2_init


!*********************************************************************
!* initial microstructural state                                     *
!*********************************************************************
pure function constitutive_j2_stateInit(myInstance)
  
 implicit none
 integer(pInt), intent(in) :: myInstance
 real(pReal), dimension(1) :: constitutive_j2_stateInit
  
 constitutive_j2_stateInit = constitutive_j2_tau0(myInstance)

end function constitutive_j2_stateInit


!*********************************************************************
!* relevant microstructural state                                    *
!*********************************************************************
pure function constitutive_j2_aTolState(myInstance)

 implicit none
 !*** input variables
 integer(pInt), intent(in) ::  myInstance                      ! number specifying the current instance of the plasticity

 !*** output variables
 real(pReal), dimension(constitutive_j2_sizeState(myInstance)) :: &
                              constitutive_j2_aTolState       ! relevant state values for the current instance of this plasticity

 constitutive_j2_aTolState = constitutive_j2_aTolResistance(myInstance)

end function constitutive_j2_aTolState


pure function constitutive_j2_homogenizedC(state,ipc,ip,el)
!*********************************************************************
!* homogenized elacticity matrix                                     *
!* INPUT:                                                            *
!*  - state           : state variables                              *
!*  - ipc             : component-ID of current integration point    *
!*  - ip              : current integration point                    *
!*  - el              : current element                              *
!*********************************************************************
 use prec, only: p_vec
 use mesh, only: mesh_NcpElems,mesh_maxNips
 use material, only: homogenization_maxNgrains,material_phase, phase_plasticityInstance
 
 implicit none
 integer(pInt), intent(in) :: ipc,ip,el
 type(p_vec), dimension(homogenization_maxNgrains,mesh_maxNips,mesh_NcpElems), intent(in) :: state
 integer(pInt) :: matID
 real(pReal), dimension(6,6) :: constitutive_j2_homogenizedC
 
 matID = phase_plasticityInstance(material_phase(ipc,ip,el))
 constitutive_j2_homogenizedC = constitutive_j2_Cslip_66(1:6,1:6,matID)

end function constitutive_j2_homogenizedC


pure subroutine constitutive_j2_microstructure(Temperature,state,ipc,ip,el)
!*********************************************************************
!* calculate derived quantities from state (not used here)           *
!* INPUT:                                                            *
!*  - Tp              : temperature                                  *
!*  - ipc             : component-ID of current integration point    *
!*  - ip              : current integration point                    *
!*  - el              : current element                              *
!*********************************************************************
 use prec, only: p_vec
 use mesh, only: mesh_NcpElems,mesh_maxNips
 use material, only: homogenization_maxNgrains,material_phase, phase_plasticityInstance
 
 implicit none
!* Definition of variables
 integer(pInt), intent(in) :: ipc,ip,el
 real(pReal), intent(in) :: Temperature
 type(p_vec), dimension(homogenization_maxNgrains,mesh_maxNips,mesh_NcpElems), intent(in) :: state
 integer(pInt) :: matID

 matID = phase_plasticityInstance(material_phase(ipc,ip,el))

end subroutine constitutive_j2_microstructure


!****************************************************************
!* calculates plastic velocity gradient and its tangent         *
!****************************************************************
pure subroutine constitutive_j2_LpAndItsTangent(Lp, dLp_dTstar_99, Tstar_v, Temperature, state, g, ip, el)

  !*** variables and functions from other modules ***!
  use prec,     only: p_vec
  use math,     only: math_mul6x6, &
                      math_Mandel6to33, &
                      math_Plain3333to99, &
                      math_deviatoric33, &
                      math_mul33xx33
  use mesh,     only: mesh_NcpElems, &
                      mesh_maxNips
  use material, only: homogenization_maxNgrains, &
                      material_phase, &
                      phase_plasticityInstance

  implicit none
  !*** input variables ***!
  real(pReal), dimension(6), intent(in)::       Tstar_v                   ! 2nd Piola Kirchhoff stress tensor in Mandel notation
  real(pReal), intent(in)::                     Temperature
  integer(pInt), intent(in)::                   g, &                      ! grain number
                                                ip, &                     ! integration point number
                                                el                        ! element number
  type(p_vec), dimension(homogenization_maxNgrains,mesh_maxNips,mesh_NcpElems), intent(in):: state ! state of the current microstructure

  !*** output variables ***!
  real(pReal), dimension(3,3), intent(out) ::   Lp                        ! plastic velocity gradient
  real(pReal), dimension(9,9), intent(out) ::   dLp_dTstar_99             ! derivative of Lp with respect to Tstar (9x9 matrix)

  !*** local variables ***!
  real(pReal), dimension(3,3) ::                Tstar_dev_33              ! deviatoric part of the 2nd Piola Kirchhoff stress tensor as 2nd order tensor
  real(pReal), dimension(3,3,3,3) ::            dLp_dTstar_3333           ! derivative of Lp with respect to Tstar as 4th order tensor
  real(pReal)                                   gamma_dot, &              ! strainrate
                                                norm_Tstar_dev, &         ! euclidean norm of Tstar_dev
                                                squarenorm_Tstar_dev      ! square of the euclidean norm of Tstar_dev
  integer(pInt)                                 matID, &
                                                k, &
                                                l, &
                                                m, &
                                                n
 
  matID = phase_plasticityInstance(material_phase(g,ip,el))

  ! deviatoric part of 2nd Piola-Kirchhoff stress
  Tstar_dev_33 = math_deviatoric33(math_Mandel6to33(Tstar_v))

  squarenorm_Tstar_dev = math_mul33xx33(Tstar_dev_33,Tstar_dev_33)
  norm_Tstar_dev = sqrt(squarenorm_Tstar_dev) 

  ! Initialization of Lp and dLp_dTstar
  Lp = 0.0_pReal
  dLp_dTstar_99 = 0.0_pReal

  ! Tstar == 0 --> both Lp and dLp_dTstar are zero
  if (norm_Tstar_dev > 0.0_pReal) then
   
    ! Calculation of gamma_dot
    gamma_dot = constitutive_j2_gdot0(matID) * (            sqrt(1.5_pReal) * norm_Tstar_dev & 
                                               / &!---------------------------------------------------
                                                 (constitutive_j2_fTaylor(matID) * state(g,ip,el)%p(1)) ) **constitutive_j2_n(matID)

    ! Calculation of Lp
    Lp = Tstar_dev_33/norm_Tstar_dev * gamma_dot/constitutive_j2_fTaylor(matID)

    !* Calculation of the tangent of Lp
    forall (k=1_pInt:3_pInt,l=1_pInt:3_pInt,m=1_pInt:3_pInt,n=1_pInt:3_pInt) &
      dLp_dTstar_3333(k,l,m,n) = (constitutive_j2_n(matID)-1.0_pReal) * Tstar_dev_33(k,l)*Tstar_dev_33(m,n) / squarenorm_Tstar_dev
    forall (k=1_pInt:3_pInt,l=1_pInt:3_pInt) &
      dLp_dTstar_3333(k,l,k,l) = dLp_dTstar_3333(k,l,k,l) + 1.0_pReal
    dLp_dTstar_99 = math_Plain3333to99(gamma_dot / constitutive_j2_fTaylor(matID) * dLp_dTstar_3333 / norm_Tstar_dev)
  end if

end subroutine constitutive_j2_LpAndItsTangent


!****************************************************************
!* calculates the rate of change of microstructure              *
!****************************************************************
pure function constitutive_j2_dotState(Tstar_v, Temperature, state, g, ip, el)

  use prec, only: &
    p_vec
  use math, only: &
    math_mul6x6
  use mesh, only: &
    mesh_NcpElems, &
    mesh_maxNips
  use material, only: &
    homogenization_maxNgrains, &
    material_phase, &
    phase_plasticityInstance
  
  implicit none
  !*** input variables ***!
  real(pReal), dimension(6), intent(in) ::  Tstar_v                   ! 2nd Piola Kirchhoff stress tensor in Mandel notation
  real(pReal), intent(in) ::                Temperature
  integer(pInt), intent(in)::               g, &                      ! grain number
                                            ip, &                     ! integration point number
                                            el                        ! element number
  type(p_vec), dimension(homogenization_maxNgrains,mesh_maxNips,mesh_NcpElems), intent(in) :: state ! state of the current microstructure
  
  !*** output variables ***!
  real(pReal), dimension(1) ::              constitutive_j2_dotState  ! evolution of state variable
  
  !*** local variables ***!
  real(pReal), dimension(6) ::              Tstar_dev_v               ! deviatoric part of the 2nd Piola Kirchhoff stress tensor in Mandel notation
  real(pReal)                               gamma_dot, &              ! strainrate
                                            hardening, &              ! hardening coefficient
                                            saturation, &             ! saturation resistance
                                            norm_Tstar_dev            ! euclidean norm of Tstar_dev
  integer(pInt)                             matID

  matID = phase_plasticityInstance(material_phase(g,ip,el))

  ! deviatoric part of 2nd Piola-Kirchhoff stress
  Tstar_dev_v(1:3) = Tstar_v(1:3) - sum(Tstar_v(1:3))/3.0_pReal
  Tstar_dev_v(4:6) = Tstar_v(4:6)
  
  norm_Tstar_dev = sqrt(math_mul6x6(Tstar_dev_v,Tstar_dev_v))
  
  ! gamma_dot
  gamma_dot = constitutive_j2_gdot0(matID) * (            sqrt(1.5_pReal) * norm_Tstar_dev & 
                                             / &!---------------------------------------------------
                                               (constitutive_j2_fTaylor(matID) * state(g,ip,el)%p(1)) ) ** constitutive_j2_n(matID)
  
  ! hardening coefficient
  if (abs(gamma_dot) > 1e-12_pReal) then
    if (constitutive_j2_tausat_SinhFitA(matID) == 0.0_pReal) then
      saturation = constitutive_j2_tausat(matID)
    else
      saturation = (  constitutive_j2_tausat(matID) &
                    + ( log(  ( gamma_dot / constitutive_j2_tausat_SinhFitA(matID)&
                                )**(1.0_pReal / constitutive_j2_tausat_SinhFitD(matID))&
                             + sqrt(  ( gamma_dot / constitutive_j2_tausat_SinhFitA(matID) &
                                       )**(2.0_pReal / constitutive_j2_tausat_SinhFitD(matID)) &
                                    + 1.0_pReal ) &
                             ) & ! asinh(K) = ln(K + sqrt(K^2 +1))
                        )**(1.0_pReal / constitutive_j2_tausat_SinhFitC(matID)) &
                    / (  constitutive_j2_tausat_SinhFitB(matID) &
                       * (gamma_dot / constitutive_j2_gdot0(matID))**(1.0_pReal / constitutive_j2_n(matID)) &
                       ) &
                    )
    endif
    hardening = ( constitutive_j2_h0(matID) + constitutive_j2_h0_slopeLnRate(matID) * log(gamma_dot) ) &
                * abs( 1.0_pReal - state(g,ip,el)%p(1)/saturation )**constitutive_j2_a(matID) &
                * sign(1.0_pReal, 1.0_pReal - state(g,ip,el)%p(1)/saturation)
  else
    hardening = 0.0_pReal
  endif
  
  ! dotState
  constitutive_j2_dotState = hardening * gamma_dot

end function constitutive_j2_dotState



!*********************************************************************
!* (instantaneous) incremental change of microstructure              *
!*********************************************************************
function constitutive_j2_deltaState(Tstar_v, Temperature, state, g,ip,el)

use prec,     only: pReal, &
                    pInt, &
                    p_vec
use mesh,     only: mesh_NcpElems, &
                    mesh_maxNips
use material, only: homogenization_maxNgrains, &
                    material_phase, &
                    phase_plasticityInstance

implicit none

!*** input variables
integer(pInt), intent(in) ::                g, &                      ! current grain number
                                            ip, &                     ! current integration point
                                            el                        ! current element number
real(pReal), intent(in) ::                  Temperature               ! temperature
real(pReal), dimension(6), intent(in) ::    Tstar_v                   ! current 2nd Piola-Kirchhoff stress in Mandel notation
type(p_vec), dimension(homogenization_maxNgrains,mesh_maxNips,mesh_NcpElems), intent(in) :: &
                                            state                     ! current microstructural state

!*** output variables
real(pReal), dimension(constitutive_j2_sizeDotState(phase_plasticityInstance(material_phase(g,ip,el)))) :: &
                                            constitutive_j2_deltaState ! change of state variables / microstructure
 
!*** local variables


constitutive_j2_deltaState = 0.0_pReal

endfunction


!****************************************************************
!* calculates the rate of change of temperature                 *
!****************************************************************
pure function constitutive_j2_dotTemperature(Tstar_v, Temperature, state, g, ip, el)

  !*** variables and functions from other modules ***!
  use prec,     only: p_vec
  use mesh,     only: mesh_NcpElems,mesh_maxNips
  use material, only: homogenization_maxNgrains
  
  implicit none
  !*** input variables ***!
  real(pReal), dimension(6), intent(in) ::  Tstar_v                   ! 2nd Piola Kirchhoff stress tensor in Mandel notation
  real(pReal), intent(in) ::                Temperature
  integer(pInt), intent(in)::               g, &                      ! grain number
                                            ip, &                     ! integration point number
                                            el                        ! element number
  type(p_vec), dimension(homogenization_maxNgrains,mesh_maxNips,mesh_NcpElems), intent(in) :: state ! state of the current microstructure
  
  !*** output variables ***!
  real(pReal) constitutive_j2_dotTemperature                          ! rate of change of temperature
  
  ! calculate dotTemperature
  constitutive_j2_dotTemperature = 0.0_pReal

end function constitutive_j2_dotTemperature


!*********************************************************************
!* return array of constitutive results                              *
!*********************************************************************
pure function constitutive_j2_postResults(Tstar_v, Temperature, dt, state, g, ip, el)

!*** variables and functions from other modules ***!
  use prec,     only: p_vec
  use math,     only: math_mul6x6
  use mesh,     only: mesh_NcpElems, &
                      mesh_maxNips
  use material, only: homogenization_maxNgrains, &
                      material_phase, &
                      phase_plasticityInstance, &
                      phase_Noutput

  implicit none
  !*** input variables ***!
  real(pReal), dimension(6), intent(in)::   Tstar_v                    ! 2nd Piola Kirchhoff stress tensor in Mandel notation
  real(pReal), intent(in)::                 Temperature, &
                                            dt                         ! current time increment
  integer(pInt), intent(in)::               g, &                       ! grain number
                                            ip, &                      ! integration point number
                                            el                         ! element number
  type(p_vec), dimension(homogenization_maxNgrains,mesh_maxNips,mesh_NcpElems), intent(in) :: state ! state of the current microstructure
  
  !*** output variables ***!
  real(pReal), dimension(constitutive_j2_sizePostResults(phase_plasticityInstance(material_phase(g,ip,el)))) :: &
                                            constitutive_j2_postResults
  
  !*** local variables ***!
  real(pReal), dimension(6) ::              Tstar_dev_v                ! deviatoric part of the 2nd Piola Kirchhoff stress tensor in Mandel notation
  real(pReal)                               norm_Tstar_dev             ! euclidean norm of Tstar_dev
  integer(pInt)                             matID, &
                                            o, &
                                            c

  !*** global variables ***!
  ! constitutive_j2_gdot0
  ! constitutive_j2_fTaylor
  ! constitutive_j2_n
  
  
  matID = phase_plasticityInstance(material_phase(g,ip,el))
  
  ! calculate deviatoric part of 2nd Piola-Kirchhoff stress and its norm
  Tstar_dev_v(1:3) = Tstar_v(1:3) - sum(Tstar_v(1:3))/3.0_pReal
  Tstar_dev_v(4:6) = Tstar_v(4:6)
  norm_Tstar_dev = sqrt(math_mul6x6(Tstar_dev_v,Tstar_dev_v))
  
  c = 0_pInt
  constitutive_j2_postResults = 0.0_pReal

  do o = 1_pInt,phase_Noutput(material_phase(g,ip,el))
    select case(constitutive_j2_output(o,matID))
      case ('flowstress')
        constitutive_j2_postResults(c+1_pInt) = state(g,ip,el)%p(1)
        c = c + 1_pInt
      case ('strainrate')
        constitutive_j2_postResults(c+1_pInt) = &
              constitutive_j2_gdot0(matID) * (            sqrt(1.5_pReal) * norm_Tstar_dev & 
                                             / &!---------------------------------------------------
                                               (constitutive_j2_fTaylor(matID) * state(g,ip,el)%p(1)) ) ** constitutive_j2_n(matID)
        c = c + 1_pInt
    end select
  enddo

end function constitutive_j2_postResults

end module constitutive_j2